Joint Assumptions For Both Trial Arms

Why vary both arms together?

A treatment-effect estimate can depend on what is assumed about outcomes that were not observed. It is tempting to change the assumption for one arm while leaving the other fixed. That can be useful for isolating an arm’s contribution, but it does not necessarily represent the joint scenario of interest: missing outcomes may be less favorable than predicted in both arms, or the departures may differ between arms. A joint sensitivity analysis makes both assumptions explicit and evaluates the treatment contrast under their combination.

The key is to keep the two questions separate: what is assumed about the missing outcomes in each arm, and what treatment contrast follows from those assumptions? A pair of sensitivity parameters answers the first question; an assumption-indexed contrast answers the second. The parameters are not quantities estimated from the observed outcomes. They encode alternative assumptions for outcomes that are unavailable.

The meaning of “positive” depends on the outcome scale. For an outcome where larger values are better, a positive departure means that the mean among missing participants is assumed higher than its reference prediction. For an outcome where smaller values are better, the clinical interpretation reverses, even though the arithmetic does not. State the outcome scale and the direction of benefit before interpreting a scenario.

A two-arm mixture model

Consider a simple marginal-mean analysis with a continuous outcome measured in the same units in both arms. In arm \(a\), where \(a\) is \(T\) for treatment or \(C\) for control, let \(q_a\) be the fraction with observed outcomes, \(\bar{y}_{a,\mathrm{obs}}\) the observed-outcome mean, and \(m_a^0\) the reference predicted mean for participants whose outcomes are missing. The reference prediction could, for example, come from a specified reference analysis. The sensitivity parameter \(\delta_a\) shifts that missing-outcome mean from \(m_a^0\) to \(m_a^0+\delta_a\); it does not shift the observed outcomes.

This setup treats the arm’s outcome distribution as a mixture of an observed group and a missing group. The target arm mean under the chosen assumption is the observed fraction’s contribution plus the missing fraction’s contribution. The treatment contrast is defined here as treatment minus control; positive values favor treatment only when higher outcomes are better.

The reference contrast \(\hat{\theta}(0,0)\) is the contrast when both departures are zero. Each departure is weighted by its arm’s missing fraction: a one-unit change in the assumed missing-outcome mean changes that arm’s estimated overall mean by the fraction missing. The minus sign before the control term follows from defining the contrast as treatment minus control.

The formula relies on a clearly specified reference prediction in each arm, an additive shift on the stated outcome scale, and fixed observed data and response fractions across scenarios. It also assumes that each \(\delta_a\) is a shift in the mean for missing participants, not a shift in the observed-group mean. The formula is exact for this marginal mixture calculation. Other estimators, such as covariate-adjusted estimators with different standardization weights, require their own corresponding derivation; the mixture weights should not be transferred without justification.

Worked joint scenarios

Symmetry, asymmetry, and interpretation

“Symmetric” can refer to more than one scenario, so a report should state the actual parameter values rather than rely on the label alone. In an equal-departure scenario, \(\delta_T=\delta_C\): the two arms’ missing means move by the same number of outcome units from their own reference predictions. In an equal-magnitude, opposite-direction scenario, \(\delta_T=-\delta_C\): one arm is shifted above its reference by the same amount that the other is shifted below. Neither construction implies that the resulting changes in the arm means, or in the treatment contrast, have equal magnitude.

An asymmetric scenario simply assigns different departures to the two arms, such as \((-3,+1)\) points. Asymmetry may be important when there are substantive reasons to expect the arms’ missing outcomes to differ in their departures from the reference predictions. The chosen values should be described in terms of those predictions and the outcome’s units, not just as “more favorable” or “less favorable.” In particular, a zero departure in one arm can coexist with a nonzero departure in the other.

A useful check is to calculate each arm’s change before interpreting the contrast. For any pair, those changes are \((1-q_T)\delta_T\) for treatment and \((1-q_C)\delta_C\) for control. The contrast change is the treatment change minus the control change. This check catches sign errors and makes clear why two assumptions that sound similar may have different effects on the estimate.

Setting up and checking a joint analysis

A disciplined analysis specifies the assumptions before calculating scenario-specific estimates. It should identify the reference predictions, the outcome scale and direction of benefit, the meaning and units of each departure, and the joint pairs to be examined. A pair is a scenario: changing one coordinate while silently holding the other fixed answers a different question.

Diagnostics are substantive as well as arithmetic. Check that assumed missing means are possible on the outcome scale; for a bounded score, a shifted mean must remain within its feasible range. Compare departures with clinically interpretable differences and the uncertainty of the reference predictions. Check that a proposed scenario is internally coherent across arms and consistent with the reasons outcomes are missing. If the assumptions imply impossible values or lack a defensible interpretation, revise or reject the scenario rather than relying on a precise calculation.

How this approach differs from alternatives

Changing one arm’s departure at a time is useful for showing how sensitive the estimate is to that arm, but it does not capture joint departures. Constraining both parameters to be equal reduces the analysis to a one-dimensional set of assumptions; it is transparent, but excludes asymmetric possibilities. Allowing separate parameters permits those possibilities, provided the joint scenarios are selected and explained rather than chosen after seeing which result is most favorable.

A pattern-mixture analysis expresses assumptions through distributions or means within observed and missing patterns. A selection-model analysis instead expresses assumptions through the relationship between outcome and the probability of being observed. These are different parameterizations of assumptions about missingness; neither makes unobserved outcomes directly observable. The two-parameter mean-shift calculation here is a deliberately simple way to show how arm-specific assumptions enter a particular marginal contrast.

The limitations matter. The calculation does not identify the departures from observed data alone, validate the reference predictions, or account for sampling uncertainty. Nor does one pair summarize all plausible assumptions. Conclusions can depend on the outcome scale, the reference model, response fractions, and the set of joint scenarios considered. A numerical contrast under a scenario should therefore be reported as conditional on that scenario, not as an assumption-free estimate. Interval estimation for each fixed pair and conclusions across a sensitivity set are separate inferential questions.

Common mistakes

• Changing only one parameter without saying so. A result at \((\delta_T,0)\) is a joint scenario with the control departure fixed at zero, not a general two-arm assessment.
• Assuming equal shifts cancel. They cancel in the contrast only if their missing-fraction-weighted changes are equal. Different missing fractions usually prevent exact cancellation.
• Reversing the control sign. For a treatment-minus-control contrast, a positive control departure contributes \(-(1-q_C)\delta_C\).
• Calling sensitivity values estimated effects. Departures are assumptions about unobserved outcomes, not treatment effects estimated from observed data.
• Confusing assumption sensitivity with precision. A changed point estimate across scenarios reflects changed assumptions; by itself it does not quantify sampling uncertainty.